Vector algebra

[Tolaso J Kos]

Let $\mathbf{a} , \mathbf{b}, \mathbf{c}$ be three non coplanar vector. If $\displaystyle{\mathbf{a}' = \frac{\mathbf{b} \times \mathbf{c}}{\left [ \mathbf{a,b, c} \right ]} \; , \; \mathbf{b}' = \frac{\mathbf{c} \times \mathbf{a}}{\left [ \mathbf{a,b, c} \right ]} \; , \; \mathbf{c}' = \frac{\mathbf{a} \times \mathbf{b}}{\left [ \mathbf{a,b, c} \right ]}}$ then prove that:

1. $\mathbf{a} \cdot \mathbf{a} ' = \mathbf{b} \cdot \mathbf{b} ' = \mathbf{c} \cdot \mathbf{c} ' = 1$.
2. $\left [ \mathbf{a,b, c} \right ] \left [ \mathbf{a}' , \mathbf{b}' , \mathbf{c}' \right ] = 1$.